Nonparametric Neutral Network Estimation of Lyapunov Exponents and a Direct Test for Chaos

This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. A simulation study to evaluate small sample performance is reported. We also apply our procedures to daily stock return data. In most cases, the hypothesis of chaos in the stock return series is rejected at the 1% level with an exception in some higher power transformed absolute returns.

[1]  R. Gencay,et al.  Lyapunov Exponents as a Nonparametric Diagnostic for Stability Analysis , 1992 .

[2]  Ramazan Gençay,et al.  A statistical framework for testing chaotic dynamics via Lyapunov exponents , 1996 .

[3]  O. Linton,et al.  The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series , 1999 .

[4]  C. Granger,et al.  A long memory property of stock market returns and a new model , 1993 .

[5]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[6]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[7]  A. Gallant,et al.  Finding Chaos in Noisy Systems , 1992 .

[8]  A. Gallant,et al.  Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data , 1991 .

[9]  Apostolos Serletis,et al.  Random Walks, Breaking Trend Functions, and the Chaotic Structure of the Velocity of Money , 1995 .

[10]  V. N. Gabushin.  Inequalities for the norms of a function and its derivatives in metric Lp , 1967 .

[11]  Y. Makovoz.  Random Approximants and Neural Networks , 1996 .

[12]  A. Gallant,et al.  Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression , 1992 .

[13]  Richard J. Smith,et al.  Estimating local Lyapunov exponents , 1997 .

[14]  Xiaotong Shen,et al.  Sieve extremum estimates for weakly dependent data , 1998 .

[15]  William A. Barnett,et al.  Robustness of nonlinearity and chaos tests to measurement error, inference method, and sample size , 1995 .

[16]  H. Furstenberg,et al.  Products of Random Matrices , 1960 .

[17]  R. Gencay,et al.  An algorithm for the n Lyapunov exponents of an n -dimensional unknown dynamical system , 1992 .

[18]  Ramazan Gençay,et al.  The Identification of Spurious Lyapunov Exponents in Jacobian Algorithms , 1996 .

[19]  Halbert White,et al.  On learning the derivatives of an unknown mapping with multilayer feedforward networks , 1992, Neural Networks.

[20]  William A. Barnett,et al.  A single-blind controlled competition among tests for nonlinearity and chaos , 1997 .

[21]  Xavier de Luna,et al.  Characterizing the Degree of Stability of Non-linear Dynamic Models , 2001 .

[22]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[23]  Halbert White,et al.  Improved Rates and Asymptotic Normality for Nonparametric Neural Network Estimators , 1999, IEEE Trans. Inf. Theory.

[24]  William A. Barnett,et al.  A single-blind controlled competition among tests for nonlinearity and chaos , 1996 .

[25]  B. LeBaron,et al.  A test for independence based on the correlation dimension , 1996 .

[26]  B. Yegnanarayana,et al.  Artificial Neural Networks , 2004 .

[27]  Jeffrey S. Racine,et al.  Semiparametric ARX neural-network models with an application to forecasting inflation , 2001, IEEE Trans. Neural Networks.

[28]  Mototsugu Shintani,et al.  Is There Chaos in the World Economy? A Nonparametric Test Using Consistent Standard Errors , 2001 .

[29]  Abhay Abhyankar,et al.  Uncovering nonlinear structure in real-time stock-market indexes: the S&P 500, the DAX, the Nikkei 225, and the FTSE-100 , 1997 .

[30]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[31]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[32]  A. Kirman,et al.  Nonlinear Dynamics and Economics , 1997 .

[33]  Xiaotong Shen,et al.  On methods of sieves and penalization , 1997 .

[34]  D. Nychka,et al.  Local Lyapunov exponents: Predictability depends on where you are , 1995 .

[35]  Kurt Hornik,et al.  Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives , 1994, Neural Computation.

[36]  Andrew R. Barron,et al.  Universal approximation bounds for superpositions of a sigmoidal function , 1993, IEEE Trans. Inf. Theory.

[37]  Simone Giannerini,et al.  New resampling method to assess the accuracy of the maximal Lyapunov exponent estimation , 2001 .

[38]  W. Wong,et al.  Convergence Rate of Sieve Estimates , 1994 .

[39]  W. Brock,et al.  Heterogeneous beliefs and routes to chaos in a simple asset pricing model , 1998 .

[40]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[41]  N. Herrndorf.  A Functional Central Limit Theorem for Weakly Dependent Sequences of Random Variables , 1984 .

[42]  Halbert White,et al.  Artificial neural networks: an econometric perspective ∗ , 1994 .